Matt Damon said he is “absolutely sick” to his stomach over the sexual assault allegations against Hollywood mogul Harvey Weinstein, denying a report claiming he tried to pressure The New York Times into killing a piece about the accusations back in 2004.

Sharon Waxman, founder of The Wrap, said the story she wrote for The Times in 2004 detailing the allegations against Mr. Weinstein and Miramax Italy head Fabrizio Lombardo “never ran,” due to pressure by Mr. Weinstein himself and phone calls from Mr. Damon and Russell Crowe vouching for Mr. Lombardo.

Instead, the story was “stripped of any reference to sexual favors or coercion and buried on the inside of the Culture section,” Ms. Waxman wrote Sunday.

Mr. Damon, a father of four daughters, has come under increased pressure in Hollywood to denounce Mr. Weinstein’s conduct, which includes the alleged rape of three women, and address Ms. Waxman’s story.

Mr. Damon told Deadline on Tuesday that he did have a brief phone conversation with Ms. Waxman after Mr. Weinstein informed him that she was writing a “negative” story about Mr. Lombardo, but said he had no idea what the story was about or that it had anything to do with Mr. Weinstein.

“Harvey said, Sharon Waxman is writing a story about Fabrizio and it’s really negative. Can you just call and tell her what your experience with Fabrizio was. So I did, and that’s what I said to her,” Mr. Damon said. “I’m sure I mentioned to her that I didn’t know anything about the rest of her piece, because I didn’t. And I still don’t know anything about that and Fabrizio. My experience with him was all above board and that’s what I told her.

“I just remember it being a negative piece, a hit job on Fabrizio, was what Harvey was saying,” he said. “Basically, that he had no professional experience. Harvey said, you worked with him. Can you tell her that he was a professional and you had a good experience, and that was it. I didn’t mind doing it, because that was all true.

“For the record, I would never, ever, ever try to kill a story like that. I just wouldn’t do that. It’s not something I would do, for anybody,” he added.

Mr. Damon said that while he was never aware of any sexual misconduct by Mr. Weinstein during the years he’s known him, the allegations against the film producer signify “the kind of sexual predation that keeps me up at night.”

“We know this stuff goes on in the world,” Mr. Damon said. “I did five or six movies with Harvey. I never saw this. I think a lot of actors have come out and said, everybody’s saying we all knew. That’s not true. This type of predation happens behind closed doors, and out of public view. If there was ever an event that I was at and Harvey was doing this kind of thing and I didn’t see it, then I am so deeply sorry, because I would have stopped it. And I will peel my eyes back now, father than I ever have, to look for this type of behavior. Because we know that it happens. I feel horrible for these women and it’s wonderful they have this incredible courage and are standing up now.

“We can all feel this change that’s happening, which is necessary and overdue,” he said. “Men are a huge part of that change, and we have to be vigilant and we have to help protect and call this stuff out because we have our sisters and our daughters and our mothers. This kind of stuff can’t happen. This morning, I just feel absolutely sick to my stomach.”